Dispersive Hydrodynamics and Applications

海洋中的海啸和强激光光纤就是波的传播速度取决于波的波长和振幅/强度的例子。在支撑波浪的介质中，这两种特性的巧合并不局限于海洋和纤维，而是在自然界、实验室和技术中无处不在。有一个新兴的应用数学领域叫做色散流体力学，它探索这两个性质的数学和物理分支。该项目将开发各种物理环境下非线性色散波的新数学解决方案和模型。解决方案将包括孤子或孤立波和色散激波，它们是波的典型表现，其速度取决于波的振幅和波长。此外，还将建立一个内部实验室浅水实验，以实现和测试发展数学的物理含义。在流体动力学、非线性光学和量子流体方面的其他应用将是这项研究的副产品。因此，这个项目是真正的跨学科。色散流体力学已成为描述色散介质中多尺度非线性波动现象的统一数学框架。小尺度孤子/孤波和大尺度水动力波（色散激波、稀薄波）都是非线性色散波方程的解。该项目将可积和不可积色散偏微分方程、凸和非凸色散流体动力系统、多维波、渐近分析以及包括内部浅水实验在内的应用研究结合在一起。虽然孤子和色散激波的数学已经在孤立的情况下得到了广泛的发展，但本项目探索了与色散流体动力学中孤子传播相关的丰富和物理动机的新数学问题。孤子和色散流体固有的尺度分离特性，使我们能够用线性退化场的双曲型偏微分方程组，以方便和通用的形式渐近地描述它们的相互作用。将这一描述扩展并应用于非凸、多维和有吸引力的分散流体动力学流动是本项目的主要目的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Tsunamis in the ocean and intense laser fiber optics are examples of waves whose speed of propagation depend on both the wave's wavelength and amplitude/intensity. The coincidence of these two properties in media that support waves is not limited to the ocean and fibers but is ubiquitous in nature, the lab, and technology. There is an emerging field of applied mathematics called dispersive hydrodynamics that explores the mathematical and physical ramifications of these two properties. This project will develop new mathematical solutions and models of nonlinear, dispersive waves in a variety of physical environments. The solutions will include solitons or solitary waves and dispersive shock waves, quintessential manifestations of waves whose speeds depend on wave amplitude and wavelength. An in-house laboratory shallow water experiment will also be constructed to realize and test the physical implications of the developed mathematics. Additional applications to fluid dynamics, nonlinear optics, and quantum fluids will be by-products of this research. Consequently, this project is truly interdisciplinary.Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media. Small-scale solitons/solitary waves and large-scale hydrodynamic waves (dispersive shock waves, rarefaction waves) are both solutions of nonlinear dispersive wave equations. This project weaves together research on integrable and nonintegrable dispersive partial differential equations, convex and nonconvex dispersive hydrodynamic systems, multidimensional waves, asymptotic analysis, and applications that include an in-house shallow water experiment. While the mathematics of solitons and that of dispersive shock waves have been extensively developed in isolation, this project explores the rich and physically motivated set of new mathematical problems related to soliton propagation in dispersive hydrodynamic flows. The scale separation inherent to solitons and dispersive hydrodynamic flows enables the asymptotic description of their interaction in a convenient and universal form in terms of a hyperbolic system of partial differential equations with a linearly degenerate field. Extending and applying this description to nonconvex, multidimensional, and attractive dispersive hydrodynamic flows are the primary aims of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.